Simplifying Square Roots
Learn how to simplify square roots by finding and extracting perfect square factors.
Definition
- The number under the radical has no perfect square factors other than 1
- There are no fractions under the radical
- There are no radicals in the denominator
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Worked Examples
Simplify
List perfect squares less than 72
→ Perfect squares to check
Find which divide 72 evenly
\checkmark, \checkmark, \checkmark → 4, 9, and 36 are factors
Choose the largest: 36
→ Factor out 36
Apply the product rule
→ Separate the roots
Simplify the perfect square
→
Answer:
Common Mistakes
Stopping too early: writing instead of
Why it's wrong: 18 still has a perfect square factor (9). Always check if the remaining radicand can be simplified further.
Correct: Continue simplifying:
Writing instead of
Why it's wrong: We take the SQUARE ROOT of 25, not 25 itself. , not 25.
Correct:
Thinking
Why it's wrong: Square roots do NOT distribute over addition! Only over multiplication.
Correct: , but . These are not equal!
Interactive Visual
Factor Tree
Enter a number to see its factor tree and prime factorization.
Click on numbers to select them. Adjust the range to explore different values.
Interactive Sandbox
Expression Calculator
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History
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Practice Problems
16 problemsWhat is ?
Why It Matters
- Exact answers: is exact, while is an approximation
- Easier calculations: Simplified forms make further operations much simpler
- Recognizing patterns: Seeing helps compare values
- Real applications: Used in geometry (diagonal of a square), physics (velocity formulas), and engineering
Real World Applications
Diagonal of a Square
Finding the diagonal of a square uses the Pythagorean theorem and requires simplifying square roots.
Example:
A square has side length 6 cm. Its diagonal is cm.
A square garden has sides of 10 meters.
What is the exact length of the diagonal path across it?
Step 1: Write the mathematical expression
Use the Pythagorean theorem:
Screen Sizes
TV and monitor sizes are measured diagonally, which involves square roots.
Example:
A monitor is 40 cm wide and 30 cm tall. The diagonal is cm.
A tablet screen is 24 cm wide and 18 cm tall.
What is the diagonal measurement?
Step 1: Write the mathematical expression
Calculate
Key Takeaways
- 1To simplify , find the largest perfect square factor of
- 2Use the product rule:
- 3Prime factorization helps find perfect square factors (pairs of primes)
- 4A square root is simplified when no perfect square factors remain under the radical
- 5Always check if your answer can be simplified further
Frequently Asked Questions
Glossary
- Radicand
- The number under the radical sign. In , the radicand is 72.
- Perfect square
- A number that is the square of an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
- Simplest form
- A square root where the radicand has no perfect square factors other than 1.
- Product rule for radicals
- for non-negative and .
Formula Card
Product Rule
Separate a square root into the product of two square roots
Simplification Pattern
Extract the perfect square factor from under the radical
Perfect Squares
Memorize these perfect squares up to 144